Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an special number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers. Note: If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.

For every $k= 1,2, \ldots$ let $s_k$ be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + \cdots + s_{200}$.

There are $ 2016 $ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $ A $ and $ B $ take turns making moves. Player $ A $ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.

Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$.

Let $ABC$ be a triangle inscribed on a circle with center $O$. Let $D$ and $E$ be points on the sides $AB$ and $BC$,respectively, such that $AD = DE = EC$. Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$. If $X \neq O$, show that, the lines $OX$ and $DE$ are perpendicular.

We say that three different integers are friendly if one of them divides the product of the other two. Let $n$ be a positive integer. a) Show that, between $n^2$ and $n^2+n$, exclusive, does not exist any triplet of friendly numbers. b) Determine if for each $n$ exists a triplet of friendly numbers between $n^2$ and $n^2+n+3\sqrt{n}$ , exclusive.